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CIS552 Indexing and Hashing 1
Indexing and Hashing
Cost estimation Basic Concepts
Ordered Indices
B+ - Tree Index Files
B - Tree Index Files
Static Hashing
Dynamic Hashing
Comparison of Ordered Indexing and Hashing
Index Definition in SQL
Multiple-Key Access
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CIS552 Indexing and Hashing 4
Index Evaluation Metrics
Access time for: Equality searchesrecords with a specified
value in an attribute
Range searchesrecords with an attribute
value falling within a specified range.
Insertion time
Deletion time
Space overhead
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CIS552 Indexing and Hashing 7
Sparse Index Files
A clustering index may be sparse.
Index records for only some search-key values.
To locate a record with search-key value kwe:
Find index record with largest search-key value < k
Search file sequentially starting at the record to which the index
record points
Less space and less maintenance overhead for insertions
and deletions.
Generally slower than dense index for locating records.
Good tradeoff: sparse index with an index entry for every
block in file, corresponding to least search-key value in the
block.
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CIS552 Indexing and Hashing 8
Example of Sparse Index Files
Brighton
Miami
Redwood
Brighton A-217 750
Downtown A-101 500
Downtown A-110 600
Miami A-215 700Perryridge A-102 400
Perryridge A-201 900
Perryridge A-218 700
Redwood A-222 700
Round Hill A-305 350
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CIS552 Indexing and Hashing 10
M
Multilevel Index (Cont.)
M
M
M
M
Index
Block 0
Data
Block 0
Index
Block 1
DataBlock 1
outer index inner index
M
M
M
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CIS552 Indexing and Hashing 11
Index Update: Deletion
If deleted record was the only record in the file with itsparticular search-key value, the search-key is deleted from
the index also.
Single-level index deletion:
Dense indicesdeletion of search-key is similar to file recorddeletion.
Sparse indicesif an entry for the search key exists in the index, it
is deleted by replacing the entry in the index with the next search-
key value in the file (in search-key order). If the next search-keyvalue already has an index entry, the entry is deleted instead of
being replaced.
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CIS552 Indexing and Hashing 13
Non-clustering Indices
Frequently, one wants to find all the records whose values
in a certain field satisfy some condition, and the file is not
ordered on the field.
Example 1: In the account database stored sequentially by account
number, we may want to find all accounts in a particular branch. Example 2: As above, but where we want to find all accounts with a
specified balance or range of balances.
We can have a non-clustering index with an index record for
each search-key value. The index record points to a bucket
that contains pointers to all the actual records with that
particular search-key value.
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CIS552 Indexing and Hashing 14
Secondary Index on balance field ofaccount
Brighton A-217 750
Downtown A-101 500
Downtown A-110 600
Miami A-215 700Perryridge A-102 400
Perryridge A-201 900
Perryridge A-218 700
Redwood A-222 700
Round Hill A-305 350
350
400
500
600
700
750
900
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CIS552 Indexing and Hashing 15
Clustering and Non-clustering
Non-clustering indices have to be dense.
Indices offer substantial benefits when searching for
records.
When a file is modified, every index on the file must be
updated. Updating indices imposes overhead on database
modification.
Sequential scan using clustering index is efficient, but a
sequential scan using a non-clustering index is expensive
each record access may fetch a new block from disk.
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CIS552 Indexing and Hashing 16
B+ - Tree Index Files
B+- Tree indices are an alternative to indexed-sequential files.
Disadvantage of indexed-sequential files: performance
degrades as file grows, since many overflow blocks get
created. Periodic reorganization of entire file is required.
Advantage of B+ - tree index files: automatically
reorganizes itself with small, local, change, in the face ofinsertions and deletions. Reorganization of entire file is not
required to maintain performance.
Disadvantage of B+-tree: extra insertion and deletion
overhead, space overhead. Advantages of B+-trees outweigh disadvantages, and they
are used extensively.
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CIS552 Indexing and Hashing 18
Typical node
Ki are the search-key values
Pi are pointers to children (for non-leaf nodes) orpointers to records or buckets of records (for leaf
nodes).
The search-keys in a node are ordered
K1 < K2 < K3< < Kn-1
B+-Tree Node Structure
P1 K1 P2 Pn-1 Kn-1 Pn
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CIS552 Indexing and Hashing 19
Leaf Nodes in B+-Trees
Properties of a leaf node: For i = 1,2,, n-1, pointer Pi either points to a file record with search-
key value Ki, or to a bucket of pointers to file records, each recordhaving search-key value Ki. Only need bucket structure if search-keydoes not form a superkey and the index is non-clustering.
If Li, Lj are leaf nodes and i < j, Lis search-key values are less than
Ljs search-key values Pn points to next leaf node in search-key order
Brighton Downtown
Brighton A-212 750
Downtown A-101 500
Downtown A-110 600
M
leaf node
account file
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CIS552 Indexing and Hashing 20
Non-Leaf Nodes in B+-Trees
Non leaf nodes form a multi-level sparse index on the leaf
nodes. For a non-leaf node with m pointers:
All the search-keys in the subtree to which Pi points are
less than Ki All the search-keys in the subtree to which Pi points are
greater than or equal to Ki1
P1 K1 P2 Pn-1 Kn-1 Pn
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CIS552 Indexing and Hashing 22
Example of a B+-tree
Leaf nodes must have between 2 and 4 values ( (n1)/2and n 1, with n=5).
Non-leaf nodes other than root must have between 3 and 5
children ( n/2 and n with n = 5). Root must have at least 2 children
Perryridge
Perryridge Redwood Round HillBrighton Downtown Miami
B+-tree for account file (n=5)
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CIS552 Indexing and Hashing 23
Observations about B+ -trees
Since the inter-node connections are done by pointers,there is no assumption that in the B+-tree, the logically
close blocks are physically close.
The non-leaf levels of the B+-tree form a hierarchy of
sparse indices. The B+-tree contains a relatively small number of levels
(logarithmic in the size of the main file), thus searches can
be conducted efficiently.
Insertions and deletions to the main file can be handledefficiently, as the index can be restructured in logarithmic
time (as we shall see).
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CIS552 Indexing and Hashing 24
Queries on B+-Trees
Find all records with a search-key value of k. Start with the root node
Examine the node for the smallest search-key value > k.
If such a value exists, assume it is Ki. Then follow Pi to the childnode
Otherwise k Km-1, where there are m pointers in the node, Thenfollow Pm to the child node.
If the node reached by following the pointer above is not a leafnode, repeat the above procedure on the node, and follow thecorresponding pointer.
Eventually reach a leaf node. If key Ki= k, follow pointer P
ito the
desired record or bucket. Else no record with search-key value kexists.
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CIS552 Indexing and Hashing 26
Updates on B+-Trees: Insertion
Find the leaf node in which the search-key value wouldappear
If the search-key value is already there in the leaf node,
record is added to file and if necessary pointer is inserted
into bucket.
If the search-key value is not there, then add the record to
the main file and create bucket if necessary. Then:
if there is room in the leaf node, insert (search-key value,
record/bucket pointer) pair into leaf node at appropriate position.
if there is no room in the leaf node, split it and insert (search-keyvalue, record/bucket pointer) pair as discussed in the next slide.
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CIS552 Indexing and Hashing 27
Updates on B+-Trees: Insertion (Cont.)
Splitting a node: take the n(search-key value, pointer) pairs (including
the one being inserted) in sorted order. Place the first
n/2 in the original node, and the rest in a new node.
Let the new node be p, and let k be the least key valuein p. Insert (k, p) in the parent of the node being split.
If the parent is full, split it and propagate the split
further up.
The splitting of nodes proceeds upwards till a node that is
not full is found. In the worst case the root node may be
split increasing the height of the tree by 1.
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CIS552 Indexing and Hashing 28
Updates on B+-Trees: Insertion(Cont.)
Perryridge
RedwoodMiami
Brighton Downtown Redwood Round HillMiami Perryridge
Perryridge
RedwoodDowntown Miami
Brighton Clearview Downtown Miami Perryridge Redwood Round Hill
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CIS552 Indexing and Hashing 29
Updates on B+-Trees:Deletion
Find the record to be deleted, and remove it from the mainfile and from the bucket (if present)
Remove (search-key value, pointer) from the leaf node if
there is no bucket or if the bucket has become empty
If the node has too few entries due to the removal, and theentries in the node and a sibling fit into a single node, then
Insert all the search-key values in the two nodes into a single node
(the one on the left), and delete the other node.
Delete the pair (Ki1, Pi), where Pi is the pointer to the deletednode, from its parent, recursively using the above procedure.
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CIS552 Indexing and Hashing 30
Updates on B+-Trees: Deletion
Otherwise, if the node has too few entries due to theremoval, and the entries in the node and a sibling dont fit
into a single node, then
Redistribute the pointers between the node and a sibling such that
both have at least the minimum number of entries
Update the corresponding search-key value in the parent of the
node.
The node deletions may cascade upwards till a node which
has n/2 or more pointers is found. If the root node has
only one pointer after deletion, it is deleted and the solechild becomes the root.
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CIS552 Indexing and Hashing 31
Examples of B+-Tree Deletion
The removal of the leaf node containing Downtown didnot result in its parent having too little pointers. So the
cascaded deletions stopped with the deleted leaf nodes
parent.
Perryridge
RedwoodMiami
Brighton Clearview Miami Perryridge Redwood Round Hill
Result after deleting Downtown from account
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CIS552 Indexing and Hashing 32
Examples of B+-Tree Deletion (Cont.)
The deleted Perryridge nodes parent became too small,but its sibling did not have space to accept one morepointer, so redistribution is performed. Observe that theroot nodes search-key value changes as a result.
Redwood
Miami
Brighton Clearview Miami Redwood Round HillDowntown
Downtown
Deletion of Perryridge instead of Downtown
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CIS552 Indexing and Hashing 34
B-Tree Index Files
Similar to B+-tree, but B-tree allows search-key values toappear only once; eliminates redundant storage of searchkeys.
Search keys in nonleaf nodes appear nowhere else in theB-tree; an additional pointer field for each search key in a
nonleaf node must be included. Generalized B-tree leaf node
Nonleaf nodepointers Bi are the bucket or file record
pointers.
P1 K1 P2 Pn-1 Kn-1 Pn
P1 B1 K1 P2 B2 K2 Pm-1 Bm-1 Km-1 Pm
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CIS552 Indexing and Hashing 35
B-Tree Index Files (Cont.)
Advantages of B-Tree indices: May use less tree nodes than a corresponding B+-Tree.
Sometimes possible to find search-key value before reaching leaf
node.
Disadvantages of B-Tree indices:
Only small fraction of all search-key values are found early
Non-leaf nodes are larger, so fan-out is reduced. Thus B-Trees
typically have greater depth than corresponding B+-Tree.
Insertion and deletion more complicated than in B+-Trees
Implementation is harder than B+
-Trees. Typically, advantages of B-Trees do not outweigh
disadvantages.
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CIS552 Indexing and Hashing 36
Static Hashing
A bucket is a unit of storage containing one or morerecords (a bucket is typically a disk block). In a hash file
organization we obtain the bucket of a record directly
from its search-key value using a hash function.
Hash function h is a function from the set of all search-keyvalues K to the set of all bucket addresses B.
Hash function is used to locate records for access,
insertion, and deletion.
Records with different search-key values may be mappedto the same bucket; thus entire bucket has to be searched
sequentially to locate a record.
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CIS552 Indexing and Hashing 37
Hash Functions
Worst hash function maps all search-key values to the same bucket;
this makes access time proportional to the number of search-key values
in the file.
An ideal hash function is uniform, i.e. each bucket is assigned the same
number of search-key values from the set of all possible values.
Ideal hash function is random, so each bucket will have the samenumber of records assigned to it irrespective of the actual distribution
of search-key values in the file.
Typical hash functions perform computation on the internal binary
representation of the search-key. For example, for a string search-key,
the binary representations of all the characters in the string could be
added and the sum modulo number of buckets could be returned.
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CIS552 Indexing and Hashing 38
Example of Hash File Organization
Brighton A-217 750
Round Hill A-305 350
Redwood A-222 700
Perryridge A-102 400
Perryridge A-201 900
Perryridge A-218 700
Miami A-215 700
Downtown A-101 500
Downtown A-110 600
bucket 0
bucket 1
bucket 2
bucket 8
bucket 7
bucket 6
bucket 5
bucket 4
bucket 3
bucket 9
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CIS552 Indexing and Hashing 39
Example of Hash File Organization (Cont.)
Hash file organization ofaccountfile, using branch-name
as key.
(See figure in previous slide.)
There are 10 buckets.
The binary representation of the ith character is assumed to
be the integeri.
The hash function returns the sum of the binary
representations of the characters modulo 10.
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CIS552 Indexing and Hashing 40
Handling of Bucket Overflows
Bucket overflow can occur because of Insufficient buckets
Skew in distribution of records. This can occur due to two reasons:
* multiple records have same search-key value
* chosen hash function produces non-uniform distribution of keyvalues
Although the probability of bucket overflow can bereduced, it can not be eliminated; it is handled by usingoverflow buckets.
Overflow chainingthe overflow buckets of a givenbucket are chained together in a linked list
Above scheme is called closed hashing. An alternative,called open hashing, is not suitable for databaseapplications.
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CIS552 Indexing and Hashing 41
Hash Indices
Hashing can be used not only for file organization, but alsofor index-structure creation. A hash index organizes the
search keys, with their associated record pointers, into a
hash file structure.
Hash indices are always secondary indicesif the file
itself is organized using hashing, a separate primary hash
index on it using the same search-key is unnecessary.
However, the term hash index is used to refer to both
secondary index structures and hash organized files.
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CIS552 Indexing and Hashing 42
Example of Hash Index
A-215
A-305
A-101
A-110
A-217
A-102
A-218
A-222
A-201
Brighton A-217 750
Downtown A-101 500
Downtown A-110 600
Miaimi A-215 700
Perryridge A-102 400
Perryridge A-201 900
Perryridge A-218 700
Redwood A-222 700
Round Hill A-305 350
bucket 0
bucket 1
bucket 2
bucket 5
bucket 4
bucket 6
bucket 3
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CIS552 Indexing and Hashing 43
Deficiencies of Static Hashing
In static hashing, function h maps search-key values to afixed set B of bucket addresses.
Databases grow with time. If initial number of buckets is too small,
performance will degrade due to too much overflows.
If file size at some point in the future is anticipated and number of
buckets allocated accordingly, significant amount of space will bewasted initially.
If database shrinks, again space will be wasted.
One option is periodic, re-organization of the file with a new hash
function, but it is very expensive.
These problems can be avoided by using techniques that
allow the number of buckets to be modified dynamically.
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CIS552 Indexing and Hashing 45
General Extendable Hash Structure
..00
..01
..10
..11
i
i1
i3
i2
hash suffix
bucket 1
bucket 2
bucket 3M
M
bucket address table
In this structure, i2 = i3 = i, whereas i1 = i1
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CIS552 Indexing and Hashing 46
Use of Extendable Hash Structure
Multiple entries in the bucket address table may point to asingle bucket. Each bucketj stores a value ij; all the entriesthat point to the same bucket have the same values on thelast ij bits of the hash suffix.
To locate the bucket containing search-keyKj:1. Compute h(Kj) = X2. Use the last i bits of X as a displacement into bucket
address table, and follow the pointer to appropriate bucket.
To insert a record with search-key valueKi, follow same
procedure as look-up and locate the bucket, sayj.If there is room in the bucketj insert record in the bucket.Else the bucket must be split and insertion re-attempted.(See next slide.)
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CIS552 Indexing and Hashing 47
Updates in Extendable Hash Structure
To split a bucketj when inserting record with search-key value Ki: Ifi > ij(more than one pointer to bucket j)
allocate a new bucket z, and set ij andjz to the old ij+1.
Make the second half of the bucket address table entries pointing toj topoint to z
remove and reinsert each record in bucketj. recompute new bucket forKi and insert record in the bucket (further
splitting is required if the bucket is still full)
Ifi = ij (only one pointer to bucket j) increment i and double the size of the bucket address table.
replace each entry in the table by two entries that point to the samebucket.
recompute new bucket address table entry forKi. Now i > ij, so use thefirst case above.
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CIS552 Indexing and Hashing 48
Update in Extendable Hash Structure (Cont.)
When inserting a value, if the bucket is full after several
splits (that is, i reaches some limit b) create an overflow
bucket instead of splitting bucket entry value further.
To delete a key value, locate it in its bucket and remove it.The bucket itself can be removed if it becomes empty
(with appropriate updates to the bucket address table).
Merging of buckets and decreasing bucket address table
size is also possible.
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CIS552 Indexing and Hashing 49
Use of Extendable Hash Structure: Example
branch-name
Brighton
Downtown
Miami
PerryridgeRedwood
Round Hill
h(branch-name)
0110 1101 1111 1011 0010 1100 0011 0000
0010 0011 1010 0000 1100 0110 1001 0001
1110 0111 1110 1101 1011 1111 0011 0011
1011 0001 0010 0100 1001 0011 0110 11110011 0101 1010 0110 1100 1001 1110 1110
1111 1000 0011 1111 1001 1100 0000 1101
00
bucket 1
Initial Hash Structure, Bucket size = 2
bucket address table
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CIS552 Indexing and Hashing 51
Example (3)
Brighton A-217 750
Downtown A-101 500
Downtown A-110 600
Miami A-215 700
2
2
2
1
hash suffix
bucket address table
Hash Structure after four insertions
..00
..01
..10
..11
E l (4)
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CIS552 Indexing and Hashing 52
Example (4)Brighton A-217 750
Downtown A-101 500
Downtown A-110 600
Miami A-102 400
Perryridge A-201 900
3
3
3
2
Hash Structure after all insertions
bucket address table
..000
..001
..010
..011
..100
..101
..110
..111
Round Hill A-432 500
3
Redwood A-222 700
Redwood A-722 750
2
Downtown A-611 820
3
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CIS552 Indexing and Hashing 53
Comparison of Ordered Indexing and Hashing
Issues to consider: Cost of periodic re-organization
Relative frequency of insertions and deletions
Is it desirable to optimize average access time at the
expense of worst-case access time?
Expected type of queries:
Hashing is generally better at retrieving records having
a specified value of the key.
If range queries are common, ordered indices are to be
preferred
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CIS552 Indexing and Hashing 54
Index Definition in SQL
Create an index
create index on
()
E.g.: create indexb-indexonbranch(branch-name)
Use create unique index to indirectly specify and enforcethe condition that the search key is a candidate key.
To drop an index
drop index
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CIS552 Indexing and Hashing 55
Multiple-Key Access
Use multiple indices for certain types of queries.
Example:selectaccount-number
fromaccount
wherebranch-name= Perryridge andBalance = 1000
Possible strategies for processing query using indices onsingle attributes:1. Use index on branch-name to find Perryridge records; test
balance = 1000.
2. Use index on balance to find accounts with balances of $1000;
test branch-name= Perryridge.3. Use branch-name index to find pointers to all records pertaining
to the Perryridge branch. Similarly use index on balance. Takeintersection of both sets of pointers obtained.
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CIS552 I d i d H hi 56
Indices on Multiple Attributes
Suppose we have an ordered index on combined search-key(branch-name, balance).
With the where clausewhere branch-name = Perryridge and balance = 1000the index on the combined search-key will fetch only records thatsatisfy both conditions.
Using separate indices is less efficientwe may fetchmany records (or pointers) that satisfy only one of theconditions.
Can also efficiently handlewherebranch-name= Perryridge andbalance < 1000
But cannot efficiently handlewherebranch-name< Perryridge andbalance = 1000May fetch many records that satisfy the first but not the secondcondition.